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Mirrors > Home > MPE Home > Th. List > equsalvw | Structured version Visualization version GIF version |
Description: Version of equsalv 2264 with a dv condition, and of equsal 2446 with two dv conditions, which requires fewer axioms. See also the dual form equsexvw 2090. (Contributed by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
equsalvw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsalvw | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.23v 2023 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) | |
2 | equsalvw.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | pm5.74i 260 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓)) |
4 | 3 | albii 1895 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
5 | ax6ev 2059 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
6 | 5 | a1bi 351 | . 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
7 | 1, 4, 6 | 3bitr4i 292 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1629 ∃wex 1852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 |
This theorem depends on definitions: df-bi 197 df-ex 1853 |
This theorem is referenced by: equvelv 2118 ax13lem2 2451 reu8 3554 asymref2 5652 intirr 5653 fun11 6101 bj-dvelimdv 33161 bj-dvelimdv1 33162 wl-clelv2-just 33705 undmrnresiss 38429 pm13.192 39130 |
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