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Mirrors > Home > MPE Home > Th. List > sbcbii | Structured version Visualization version GIF version |
Description: Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005.) |
Ref | Expression |
---|---|
sbcbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
sbcbii | ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓)) |
3 | 2 | sbcbidv 3850 | . 2 ⊢ (⊤ → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) |
4 | 3 | mptru 1543 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ⊤wtru 1537 [wsbc 3790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-sbc 3791 |
This theorem is referenced by: eqsbc2 3859 sbc3an 3860 sbccomlemOLD 3878 sbccom 3879 sbcrext 3881 sbcabel 3886 csbcow 3922 csbco 3923 sbcnel12g 4419 sbcne12 4420 csbcom 4425 csbnestgfw 4427 csbnestgf 4432 sbccsb 4441 sbccsb2 4442 csbab 4445 2nreu 4449 sbcssg 4525 sbcop 5499 sbcrel 5792 difopabOLD 5843 sbcfung 6591 tfinds2 7884 frpoins3xpg 8163 frpoins3xp3g 8164 mpoxopovel 8243 f1od2 32738 bnj62 34712 bnj89 34713 bnj156 34720 bnj524 34729 bnj610 34739 bnj919 34759 bnj976 34769 bnj110 34850 bnj91 34853 bnj92 34854 bnj106 34860 bnj121 34862 bnj124 34863 bnj125 34864 bnj126 34865 bnj130 34866 bnj154 34870 bnj155 34871 bnj153 34872 bnj207 34873 bnj523 34879 bnj526 34880 bnj539 34883 bnj540 34884 bnj581 34900 bnj591 34903 bnj609 34909 bnj611 34910 bnj934 34927 bnj1000 34933 bnj984 34944 bnj985v 34945 bnj985 34946 bnj1040 34964 bnj1123 34978 bnj1452 35044 bnj1463 35047 sbcalf 38100 sbcexf 38101 sbccom2lem 38110 sbccom2 38111 sbccom2f 38112 sbccom2fi 38113 csbcom2fi 38114 rspcsbnea 42112 2sbcrex 42771 2sbcrexOLD 42773 sbcrot3 42778 sbcrot5 42779 2rexfrabdioph 42783 3rexfrabdioph 42784 4rexfrabdioph 42785 6rexfrabdioph 42786 7rexfrabdioph 42787 rmydioph 43002 expdiophlem2 43010 sbcheg 43768 sbc3or 44529 trsbc 44537 onfrALTlem5 44539 eqsbc2VD 44837 sbcoreleleqVD 44856 onfrALTlem5VD 44882 ich2exprop 47395 |
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