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Mirrors > Home > MPE Home > Th. List > dfsbcq2 | Structured version Visualization version GIF version |
Description: This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 2069 and substitution for class variables df-sbc 3778. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3779. (Contributed by NM, 31-Dec-2016.) |
Ref | Expression |
---|---|
dfsbcq2 | ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2822 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
2 | df-clab 2711 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
3 | df-sbc 3778 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
4 | 3 | bicomi 223 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑) |
5 | 1, 2, 4 | 3bitr3g 313 | 1 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 [wsb 2068 ∈ wcel 2107 {cab 2710 [wsbc 3777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-clab 2711 df-cleq 2725 df-clel 2811 df-sbc 3778 |
This theorem is referenced by: sbsbc 3781 sbc8g 3785 sbc2or 3786 sbceq1a 3788 sbc5ALT 3806 sbcng 3827 sbcimg 3828 sbcan 3829 sbcor 3830 sbcbig 3831 sbcim1 3833 sbcal 3841 sbcex2 3842 sbcel1v 3848 sbctt 3853 sbcralt 3866 sbcreu 3870 rspsbc 3873 rspesbca 3875 sbcel12 4408 sbceqg 4409 csbif 4585 rexreusng 4683 sbcbr123 5202 opelopabsb 5530 csbopab 5555 csbopabgALT 5556 iota4 6522 csbiota 6534 csbriota 7378 onminex 7787 findes 7890 nn0ind-raph 12659 uzind4s 12889 nn0min 32014 sbcrexgOLD 41509 |
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