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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2sbcrex | Structured version Visualization version GIF version |
Description: Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.) |
Ref | Expression |
---|---|
2sbcrex | ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcrex 3864 | . . 3 ⊢ ([𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑) | |
2 | 1 | sbcbii 3832 | . 2 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ [𝐴 / 𝑎]∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑) |
3 | sbcrex 3864 | . 2 ⊢ ([𝐴 / 𝑎]∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑) | |
4 | 2, 3 | bitri 275 | 1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wrex 3064 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-v 3470 df-sbc 3773 |
This theorem is referenced by: 2rexfrabdioph 42094 4rexfrabdioph 42096 |
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