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Mirrors > Home > MPE Home > Th. List > sbciegf | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
sbciegf.1 | ⊢ Ⅎ𝑥𝜓 |
sbciegf.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbciegf | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbciegf.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | sbciegf.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ax-gen 1797 | . 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
4 | sbciegft 3756 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | |
5 | 1, 3, 4 | mp3an23 1450 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 [wsbc 3720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-sbc 3721 |
This theorem is referenced by: sbcieg 3758 rexsngf 4570 ralsngf 4571 opelopabgf 5392 opelopabf 5397 eqerlem 8306 bnj919 32148 bnj1464 32226 bnj1123 32368 bnj1373 32412 poimirlem25 35082 sbccomieg 39734 aomclem6 40003 fveqsb 41157 |
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